Controllability of semilinear stochastic functional integrodifferential systems in Hilbert spaces

Banach fixed-point theorem in semilinear controllability problems – a survey

“…In papers [73] and [74] the authors present sufficient conditions for the controllability of stochastic integro-differential systems in a finite-dimensional space. Two years later in paper [75] Balachandran, Park and Subalakshmi examine the complete controllability of the stochastic semilinear functional integro-differential system defined as follows…”

“…Definition 2 [75]. A stochastic process X is said to be a mild solution of the system (1) if the following conditions are satisfied:…”

“…In order to define complete controllability, we should put some hypotheses [75]: Hypothesis 1. The functions f , σ and g satisfy the following Lipschitz condition and for arbitrary γ i , ξ i ∈ C α , i = 1, 2 and 0 ≤ t ≤ T , suppose that there exist positive real constants…”

Approximate Controllability of Semilinear Neutral Stochastic Integrodifferential Inclusions with Infinite Delay

“…In fact, many deterministic models often fluctuate because of noise, so we must move from deterministic control problems to stochastic control problems. The applications of stochastic differential equations and stochastic delay differential equations have attracted great interest; see [9][10][11][12][13]. In particular, Balasubramaniam and Ntouyas [14] have studied the controllability of the following neutral stochastic functional differential inclusions with infinite delay in abstract space:…”

Complete Controllability of Impulsive Stochastic Integrodifferential Systems in Hilbert Space

“…where 0 is F 0 -measurable, is separable Hilbert space, is the infinitesimal generator of a strongly continuous semigroup ( ) on , ∈ L( , ), ( ) is feedback control, ( ) is -Wiener process, and Σ ∈ L 2 ( 1/2 , ). For nonlinear stochastic systems in infinite dimensional space, there are also many results on the controllability theory (see [8][9][10][11][12][13]). On the other hand, the impulsive effects exist widely in many evolution processes in which the states are changed abruptly at certain moments of time, involving fields such as finance, economics, mechanics, electronics, and telecommunications (see [14] and references of therein).…”